This a probably very easy question and I am not sure whether it has been asked before (although I searched for it). Moreover I really hope this is nothing which can be found in any standard commutative algebra text book.

Is there a thorough discussion of Taylor expansions of polynomials (or maybe rational functions) with coefficients in an arbitrary field?

Problem: In the classical Taylor expansion we have coefficients of the form $\frac{1}{n!}$, which obviously don't have to exists in any field, say, of finite characteristic.

Motivation: I came across it at a discussion of the Zariski tangent space to a scheme $X$ at a point $y$. Say $X=\mathbb A_k^n$ and $y\in X$ corresponds to the maximal ideal $\frak m$. Then the differential $D_y$ induces an isomorphism $\frak {m}/\frak {m}^2\rightarrow E^\vee$, where $\frak E^\vee$ is the dual of the vector space $k^n$, thus identifies the Zariski tangent space with the "classical" tangent space as we know it from the theory of manifolds.

The prove usually involves something like: $\frak {m}=(T_1-\lambda_1,..., T_n-\lambda_n)$, then use the Taylor expansion of any polynomial $P$ in $(\lambda_1,...,\lambda_n)$.

Remark: I guess this is easy to fix for polynomials, but I wonder whether everything works in general for say rational functions, or anything fancier.